- Atom (order theory)
In the mathematical field of
order theory , given two elements "a" and "b" of apartially ordered set , one says that "b" covers "a", and writes "a" <: "b" or "b" :> "a", if "a" < "b" and there is no element "c" such that "a" < "c" < "b". In other words, "b" covers "a" if "b" is greater than "a" and minimal with this property, or equivalently if "a" is smaller than "b" and maximal with this property.In a partially ordered set with
least element 0, an atom is an element that covers 0, i.e. an element that is minimal among the non-zero elements. A partially ordered set with a least element is called atomic if every non-zero element "b" > 0 has an atom "a" below it, i.e. "b" ≥ "a" :> 0.A partially ordered set is called relatively atomic (or "strongly atomic") if for all "a" < "b" there is an element "c" such that "a" <: "c" ≤ "b" or, equivalently, if every interval ["a, b"] is atomic. Every relatively atomic partially ordered set with a least element is atomic.
A partially ordered set with least element 0 is called atomistic if every element is the
least upper bound of a set of atoms. Every finite poset is relatively atomic, and every finite poset with 0 is atomic. But the linear order with three elements is not atomistic.Atoms in partially ordered sets are abstract generalizations of singletons in
set theory . Atomicity (the property of being atomic) provides an abstract generalization in the context oforder theory of the ability to select an element from a non-empty set.External links
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